On Hopkins' Picard Group Pic_2 at the prime 3

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On Hopkins’ Picard Group Pic_2 at the prime 3

Nasko Karamanov

To cite this version:

Nasko Karamanov. On Hopkins’ Picard Group Pic_2 at the prime 3. 2009. �hal-00372518�

NASKO KARAMANOV

Abstract. In this paper we calculate the algebraic Hopkins’ Picard Group P ic

at the prime p = 3, which is a subgroup of the group of isomorphism classes of invertible K(2)-local spectra i.e. of the Hopkins’ Picard Group P ic

. We use the resolution of the K(2)-local sphere introduced in [3] and the meth- ods from [5] and [7].

1. Introduction

Let C be a symmetric monoidal category with product ∧ and unit I. We say that an object X in C is invertible if there exists an object Y in C such that X ∧ Y ∼ = I.

If the collection of equivalence classes of invertible objects is a set, then the product defines a group structure on it. We denote this group by P ic(C), the Picard group of C .

For example, by [6] the homomorphism Z → P ic(S) : n 7→ S ⁿ defines an iso- morphism between the integers and the Picard group of S, the stable homotopy category.

Let K n be the category of K(n)-local spectra, where K(n) is the n-th Morava K-theory at the prime p. The unit in K n is given by L K(n) S ⁰ and the product of two K(n)-local spectra by X ∧ Y := L K(n) (X ∧ Y ) (as the ordinary smash product of two K(n)-local spectra need not be K(n)-local). Hopkins’ Picard group is the group P ic(K n ) which we denote by P ic n . The first account of it appears in [8] and the case n = 1 is treated in details in [6] where also some examples of elements of P ic 2 at the prime p = 2 are given.

In this paper we are interested in P ic 2 at the prime p = 3.

One way to study K n is through the functor E n∗ X := π ∗ L K(n) (E n ∧ X ) where E n is the Lubin-Tate (commutative ring) spectrum with coefficients ring E n∗ ∼ = WF _p

[[u 1 , . . . , u n−1 ]][u, u ⁻¹ ], where the power series ring is over the Witt vectors WF _p

. Recall that E n is acted on by the (Big) Morava Stabilizer group G _n = S _n ⋊ Gal( F _p

/ F _p ) by E ∞ -maps (Goerss-Hopkins, Hopkins-Miller). Let EG n be the category of profinite E n∗ [[ G _n ]]-modules, i.e. E n∗ -modules with an G _n -action compatible with the action of G _n on E n∗ . The tensor product (over (E n ) ∗ ) gives a monoidal structure on EG n .

Proposition 1.1. [6] Let X ∈ K n . Then the following conditions are equivalent : a) X is invertible in K n ;

b) E n∗ X is free E n ∗ -module of rank 1;

c) E n∗ X is invertible in EG n .

1.1. Let P ic ^alg _n := P ic(EG n ). By Proposition 1.1 there is a homomorphism : ǫ n : P ic n → P ic ^alg _n

X 7→ E n∗ X .

1

Let P ic ^alg,0 _n be the subgroup of P ic ^alg _n of index 2 of modules concentrated in even degrees. Let M ∈ P ic ^alg,0 _n and ι M be a generator of M in degree 0, as an (E n ) ∗ - module . Then for all g ∈ G _n there exists a unique element u g ∈ (E n ) ^× ₀ such that g ∗ (ι M ) = u g ι M . The map θ M : g 7→ u g is a crossed homomorphism and is a well defined element in H ¹ ( G _n ; (E n ) ^× ₀ ) that does not depend on ι M . Thus we have a homomorphism P ic ^alg,0 _n → H ¹ ( G _n ; (E n ) ^× ₀ ).

Proposition 1.2. [5] P ic ^alg,0 _n ∼ = H ¹ ( G _n ; (E n ) ^× ₀ ) .

Not much is known for the kernel κ n of ǫ n (cf. [8]). When n ² ≤ 2(p − 1) and n > 1 or when n = 1 and p > 2 it is known to be trivial. It is conjectured (Hopkins, [8]) that k n is a finite p-group.

The next theorem is an unpublished result of Goerss, Henn, Mahowald and Rezk.

Theorem 1.3. At the prime p = 3, κ 2 ∼ = Z /3 × Z /3 .

The next two theorems describe some known results for P ic n . Theorem 1.4. [6]

P ic 1 ∼ = Z ₂ × Z /2 × Z /4 p = 2 P ic 1 ∼ = Z _p × Z /2(p − 1) p > 2 .

The spectrum S ¹ is a generator of P ic 1 in the case p > 2. In an unpublished result and using Shimomura’s calculations of π ∗ L 2 S at primes p > 3 Hopkins shows Theorem 1.5. For primes p > 3

P ic 2 ∼ = Z ²

p × Z /2(p ² − 1) . The main result of this paper is the following theorem.

Theorem 1.6. At the prime 3

P ic ^alg ₂ ∼ = Z ²

3 × Z /16

generated by (E 2 ) ∗ S ¹ and (E 2 ) ∗ S ⁰ [det], where det is a suitable character of G ₂ . Theorem 1.3 and Theorem 1.6 imply the following theorem.

Theorem 1.7. At the prime 3

P ic 2 ∼ = Z ² ₃ × Z /3 × Z /3 × Z /16 .

1.2. This paper is organized as follows. In section 2 we recall the basic properties of the Morava stabilizer group G _n and describe some important subgroups in the case n = 2 and p = 3. We also recall the GHMR resolution of [3] and the spectral sequence of [5] and [7] that we use for the most difficult part of our calculation. In section 3 we define two elements of P ic ^alg _n that turn out to be generators in the case of P ic ^alg ₂ . In section 4 we present three short exact sequences that we use to simplify the calculations. In Section 5 we describe the part of the first page of the spectral sequence that is needed for the calculations. The final calculations for P ic ^alg,0 ₂ are done in section 6, and P ic ^alg ₂ is treated in the last section.

The author would like to thank Hans-Werner Henn for many usefull discussions

and for sharing his knowledge on the subject.

2. On the Morava stabilizer group and the GHMR resolution In this section we recall some basic properties of the Morava stabilizer group and some important finite subgroups in the case n = 2 and p = 3. We also describe the main tool of this work, that is the algebraic GHMR resolution of the K(2)-local sphere constructed in [3]. This resolution is used in [5] and [7] to determine the homotopy of the mod-3 Moore spectrum localized at K(2). We will use some of the calculations of [5] and [7] and the spectral sequence used there. For more details the reader is referred to the corresponding papers.

2.1. Recall that S _n is the group of automorphisms of the Honda formal group law Γ n with p-series [p] Γ

(x) = x ^p

, that is, the group of units in the endomorphism ring End(Γ n ). Let O n be the non-commutative ring extension of WF _p

(the Witt vectors over F _p

, that we denote by W from now on) generated by an element S satisfying S ⁿ = p and Sw = w ^σ S where w ∈ W and σ is the lift of the Frobenius automorphism of F _p

. Then End(Γ n ) can be identified with O n . For example, in the case n = 2 and p = 3 each element g of S ₂ can be written as g = g 1 + g 2 S with g 1 ∈ W ^× and g 2 ∈ W .

2.2. Right multiplication of S _n on End(Γ n ) defines a homomorphism S _n → GL( W ).

Composition with the determinant can be extended to G _n to obtain a homo- morphism G _n → W ^× ⋊ Gal( F _p

/ F _p ) and it is easy to check that this lands in Z ^×

p ⋊ Gal( F _p

/ F _p ). The quotient of Z ^×

p with its torsion subgroup, isomorphic to Z /(p − 1), can be identified with Z _p and we get a homomorphism called reduced determinant or reduced norm:

G _n → Z _p .

The kernel of this homomorphism is denoted by G ¹ _n and in the case when p − 1 divides n we have G _n ∼ = G ¹ _n × Z _p .

2.3. The element S generates a two sided maximal ideal m in O n with quotient O n / m ∼ = F _p

. The strict Morava stabilizer group S n is the kernel of the homomor- phism O ^× _n → F ^×

p

induced by reduction modulo m . We denote its intersection with G ¹

n by S _n ¹ .

2.4. Let n = 2 and p = 3 from now on. Let ω be a primitive eighth root of unity, φ ∈ Gal( F ₉ / F ₃ ) the generator, t := ω ² , ψ := ωφ and a := ¹ ₂ (1 + ωS). It is easy to verify that a is an element of order 3 . These elements satisfy ψa = aψ, tψ = ψt ³ , ta = a ² t and ψ ² = t ² . Then a, ψ and t generate a subgroup of order 24, denoted G 24 , ω and φ a subgroup isomorphic to the semi-dihedral group of order 16, denoted SD 16 . The elements ω ² and φ generate a subgroup of SD 16 isomorphic to the quaternion group of order 8, denoted Q 8 and we have

(1) SD 16 ∼ = Q 8 ⋊ Gal( F ₉ / F ₃ ) .

2.5. The action of the element a on F ₉ [[u 1 ]][u, u ⁻¹ ] is described in [5, Cor. 4.7]. For our purposes we only need the following formulae :

a ∗ u ≡ (1 + (1 + ω ² )u 1 )u mod (u ³ ₁ ) a ∗ u 1 ≡ u 1 − (1 + ω ² )u ² ₁ mod (u ³ ₁ ) . The (integral) action of ω is given by

(2) ω ∗ u 1 = ω ² u 1 and ω ∗ u = ωu

and the Frobenius φ acts Z ₃ -linearly by extending the action of the Frobenius on W via

(3) φ ∗ u 1 = u 1 and φ ∗ u = u .

2.6. The GHMR resolution. In [3] a resolution of the trivial G ¹ ₂ -module Z ₃ is constructed that has the following form

0 −→ C 3 ∂

−→ C 2 ∂

−→ C 1 ∂

−→ C 0 ∂

−→ Z ₃ −→ 0 where C 0 = C 3 = Z ₃ [[ G ¹

2 ]] ⊗

[G

] Z ₃ and C 1 = C 2 = Z ₃ [[ G ¹

2 ]] ⊗

[SD

] χ and χ is the non trivial character of SD 16 defined over Z ₃ , on which ω and φ act by multiplication by −1. The complete ring Z _p [[G]] is by definition lim U,n Z _p /p ⁿ [G/U ] where U runs through the open subgroups of G. Then we have the following lemma (cf. [5, Lemma 6.1]).

Lemma 2.1. Let M be a left G ¹

2 -module. Then there is a first quadrant cohomo- logical spectral sequence E _r ^∗,∗ , r ≥ 1 with

(4) E ₁ ^s,t = Ext ^t

[[

2

]] (C s ; M ) = ⇒ H ^s+t ( G ¹ ₂ ; M )

in which E ₁ ^s,t = 0 for 0 < s < 3 and t > 0, and for s ≥ 0 and t > 3, and also E ₁ ^0,t ∼ = E ^3,t ₁ ∼ = H ^t (G 24 ; M ) and E ₁ ^1,0 ∼ = E ₁ ^2,0 ∼ = Hom SD

(χ; M ) . Note that Hom SD

(χ; M ) ∼ = {m ∈ M | ω ∗ m = φ ∗ m = −m}.

2.7. Let N 0 be the kernel of ∂ 0 and j : N 0 → C 0 the inclusion. As explained in the remark after [5, Lemma 6.1] the differentials in the spectral sequence can be evaluated if we know projective resolutions Q • of N 0 and P • of C 0 as well as a chain map φ : Q • → P • covering j. These data can be assembled in a double complex T ••

with T •0 = P • , T •1 = Q • , vertical differentials δ P and δ Q and horizontal differentials (−1) ⁿ φ n : Q n → P n . The filtration of the spectral sequence of this double complex agrees (up to reindexing) with that of the spectral sequence of the lemma. Hence extension problems in the spectral sequence (4) can be studied by using the double complex. As in [5] we obtain a resolution P • := Z ₃ [[ G ¹ ₂ ]] ⊗

[G

] P _• ^′ induced from an explicit resolution of the trivial G 24 -module Z ₃ .

Lemma 2.2. [5, Lemma 6.2] Let χ be the Z ₃ [Q 8 ]-module whose underlying Z ₃ - module is Z ₃ and on which t acts by multiplication by −1 and ψ by the identity.

Then the trivial Z ₃ [G 24 ]-module Z ₃ admits a projective resolution P _• ^′ of period 4 of the following form

a

−a

−→ χ ↑ ^G _Q

^e+a+a

2

−→ χ ↑ ^G _Q

^a

2

−a

−→ χ ↑ ^G _Q

^e+a+a

2

−→ χ ↑ ^G _Q

^a

2

−a

−→ 1 ↑ ^G _Q

−→ Z ₃ .

We obtain Q • from splicing the exact complex 0 → C 3 → C 2 → C 1 → N 0 → 0

with the projective resolution P • of C 3 = C 0 (as C 1 and C 2 are projective). If we

denote by e the unit of G ¹ ₂ , by e i the generators e ⊗1 of C i and by e e i the generators

e ⊗ 1 of P i , then by [5, Lemma 6.3] there is a chain map φ : Q • → P • covering the

homomorphism j such that φ 0 : Q 0 = C 1 → P 0 sends e 1 to (e − ω) e e 0 .

2.8. We denote by E the spectral sequence for M = E 2 ∗ /(3) ∼ = F ₉ [[u 1 ]][u, u ⁻¹ ]. The structure of the E 1 -page is well known (cf. [4] for the group S _n , the case of G _n can be deduced in the same way).

Proposition 2.3. As F ₃ [β, v 1 ]-modules (β acting trivially on E ₁ ^s,∗,∗ if s = 1, 2) E ₁ ^s,∗,∗ ∼ =

 



F ₃ [[v ⁶ ₁ ∆ ⁻¹ ]][∆ ^±1 , v 1 , α, α, β]/(α e ² , α e ² , v 1 α, v 1 α, α e α e + v 1 β)e s s = 0, 3 ω ² u ⁴ F ₃ [[u ⁴ ₁ ]][v 1 , u ^±8 ]e s s = 1, 2

0 s > 3

where

|e s | = (s, 0, 0) |v 1 | = (0, 0, 4) |∆| = (0, 0, 24)

|β| = (0, 2, 12) |α| = (0, 1, 4) | α| e = (0, 1, 12) .

Recall that v 1 = u 1 u ⁻² is invariant modulo 3 with respect to the action of G ₂ and therefore all the differentials in the spectral sequence are v 1 -linear. The element α ∈ H ¹ (G 24 ; (E 2 ) 4 ) is defined as the modulo 3 reduction of δ ⁰ (v 1 ), where δ ⁰ is the Bockstein with respect to the short exact sequence

0 −→ E 2∗ −→ E 2∗ −→ E 2∗ /(3) −→ 0 .

The element α e ∈ H ¹ (G 24 ; (E 2 ) 12 ) is defined as δ ¹ (v 2 ), where v 2 = u ⁻⁸ and δ ¹ is the Bockstein with respect to the short exact sequence

0 −→ E 2∗ /(3) −→ E 2∗ /(3) −→ E 2∗ /(3, u 1 ) −→ 0

and β ∈ H ² (G 24 ; (E 2 ) 12 ) is the modulo 3 reduction of δ ⁰ δ ¹ (v 2 ). The definition of

∆ is more complicated and we have the following formula (cf. [5, Prop. 5.1]).

(5) ∆ ≡ ω ² (1 − ω ² u ² ₁ + u ⁴ ₁ )u ⁻¹² mod (u ⁶ ₁ ) .

One of the main results in [7] (see also [5, Thm. 1.2]) is the following theorem.

Theorem 2.4. There are elements

∆ k ∈ E ₁ ^0,0,24k b 2k+1 ∈ E 1,0,8(2k+1)

1 b 2k+1 ∈ E 2,0,8(2k+1) 1

for each k ∈ Z satisfying

∆ k ≡ ∆ ^k e 0 b 2k+1 ≡ ω ² u ^−4(2k+1) e 1 b 2k+1 ≡ ω ² u ^−4(2k+1) e 2

(where the first congruence is modulo (u ² ₁ ) and the last two modulo (u ⁴ ₁ )) such that

d 1 (∆ k ) =

b 2(3m+1)+1 ≡ (−1) ^m+1 ω ² (1 + u ⁴ ₁ )u ^−12k k = 2m + 1

(−1) ^m+1 mv ₁ ^4·3

⁻² b 2·3

(3m−1)+1 k = 2 · 3 ⁿ m, m 6≡ 0 mod (3)

d 1 (b 2k+1 ) =

 



(−1) ⁿ v ^6·3 ₁

⁺² b 3

(6m+1) k = 3 ⁿ⁺¹ (3m + 1) (−1) ⁿ v ^10·3 ₁

⁺² b 3

(18m+11) k = 3 ⁿ (9m + 8)

0 otherwise.

3. Two elements of P ic ^alg,0 _n

3.1. We have two distinguished elements in P ic ^alg,0 _n ∼ = H ¹ ( G _n ; (E n ) ^× ₀ ). In the case n = 2 and p = 3 these will generate the first cohomology. The first one is given by the crossed homomorphism :

η : G _n → (E n ) ^× ₀ g 7→ g ∗ u

u .

The second one is given as the composition of the reduced norm and the canonical inclusion :

det : G _n → Z ^×

p → (E n ) ^× ₀ .

We denote the corresponding elements of H ¹ ( G _n ; (E n ) ^× ₀ ) again by η and det. Note that by the isomorphism of Proposition 1.2 the element (E n ) ∗ S ² ∈ P ic ^alg,0 _n is sent to η (as u ∈ (E n ) −2 gives rise to a generator of (E n ) 0 S ² ) .

3.2. The reduction W [[u 1 ]] ^× → W ^× is equivariant with respect to the inclusion W ^× → G _n . Taking into account the Galois group we obtain a homomorphism :

red : H ¹ ( G _n ; (E n ) ^× ₀ ) → H ¹ ( W ^× ⋊ Gal; W ^× ) → H ¹ ( W ^× ; W ^× ) ^Gal

and the last homomorphism is induced by the short exact sequence 1 → W ^× → W ^× ⋊ Gal → Gal → 1 and the corresponding spectral sequence.

Proposition 3.1. Let n = 2 and p > 2. Then the image of the homomorphism red is (topologically) generated by the images of η and det.

Proof. Recall that when p > 2 then W ^× ∼ = W × F _p

with the obvious Galois action.

Thus

H ¹ ( W ^× ; W ^× ) ^Gal ∼ = End( W ^× ) ^Gal ∼ = Z ⁿ

p × Z /(p ⁿ − 1).

The image of det is given by the composition W ^× → G _n ^det → Z ^×

p → (E n ) ^× ₀ → W ^× . If g = g 0 + g 1 S + · · · + g n−1 S ⁿ⁻¹ with g i ∈ W and w ∈ W then

gw = (g 0 + g 1 S + · · · + g n−1 S ⁿ⁻¹ )w = g 0 w + g 1 w ^φ S + · · · g n−1 w ^φ

S ⁿ⁻¹ and the compostion above sends w to ww ^φ . . . w ^φ

. The image of η is given by the composition

W ^× → G _n → ^η (E n ) ^× ₀ → W ^×

and this is easily verified to be the identity.

4. Reductions

In this short section we present three short exact sequences that we use in our calculations. The last two were also used by Hopkins in the case n = 2 and p > 3.

4.1. The first one

(6) 1 → G ¹ ₂ → G ₂ → Z ₃ → 1

was described in section 2. We use the Lyndon-Hochschild-Serre spectral sequence

associated to (6) to calculate H ¹ ( G ₂ ; W [[u 1 ]] ^× ). The main difficulty is computing

H ¹ ( G ¹ ₂ ; W [[u 1 ]] ^× ). This is done in Theorem 6.4.

4.2. The reduction modulo 3 gives a short exact sequence (7) 0 → W [[u 1 ]] ^exp(p−) −→ W [[u 1 ]] ^× −→ F ₉ [[u 1 ]] ^× → 0 We will use the long exact sequence associated to (7)to calculate H ¹ ( G ¹

2 ; W [[u 1 ]]) . The difficult part is H ¹ ( G ¹

2 ; F ₉ [[u 1 ]] ^× ) (Corollary 6.2).

4.3. We have another short exact sequence coming from the reduction modulo u 1

(8) 1 → U 1 → F ₉ [[u 1 ]] ^× → F ^× ₉ → 1

where U 1 := {h ∈ F ₉ [[u 1 ]] ^× | h ≡ 1 mod (u 1 )}. The hard part is to calculate H ¹ ( G ¹

2 ; U 1 ). This is by far the hardest part of this work (cf. Theorem 5.10).

Note that the group U 1 is 3-profinite.

5. The spectral sequence

We use the spectral sequence (4) with M = U 1 and denote it by E to distinguish it from the (additive) case M = E 2∗ /(3) that we also make use of. We start with the E 1 -page. As we only need to calculate the first cohomology, it is sufficient to determine E ^0,0 ₁ , E ^0,1 ₁ and E ^1,0 ₁ ∼ = E ^2,0 ₁ and the corresponding differentials and extension problems.

5.1. The term E ^0,1 ₁ .

Proposition 5.1. H ¹ (G 24 ; F ₉ [[u 1 ]] ^× ) ∼ = Z /6.

Proof. Let F ₉ ((u 1 )) ^× be the multiplicative group of the field of fractions of F ₉ [[u 1 ]].

Each element of F ₉ ((u 1 )) ^× is of the form u ⁿ ₁ f with f ∈ F ₉ [[u 1 ]] ^× and n ∈ Z . The map

F ₉ ((u 1 )) ^× → Z : u ⁿ ₁ f 7→ n

is a group homomorphism with kernel F ₉ [[u 1 ]] ^× . Thus we have a short exact se- quence of G 24 -modules

(9) 1 → F ₉ [[u 1 ]] ^× → F ₉ ((u 1 )) ^× → Z → 1 where G 24 acts trivially on Z .

By Hilbert 90, the multiplicative version, we have H ¹ (G 24 ; F ₉ ((u 1 )) ^× ) = 0 and thus the long exact sequence induced by (9) yields

H ⁰ (G 24 ; F ₉ [[u 1 ]] ^× ) → H ⁰ (G 24 ; F ₉ ((u 1 )) ^× ) → H ⁰ (G 24 ; Z ) ։ H ¹ (G 24 ; F ₉ [[u 1 ]] ^× ).

By Proposition 2.3 we have H ⁰ (G 24 ; F ₉ [[u 1 ]] ^× ) ∼ = F ₃ [[v ⁶ ₁ ∆ ⁻¹ ]] ^× and by a simi- lar argument we conclude H ⁰ (G 24 ; F ₉ ((u 1 )) ^× ) ∼ = F ₃ ((v ⁶ ₁ ∆ ⁻¹ )) ^× . By (5) we have v ₁ ⁶ ∆ ⁻¹ ≡ u ⁶ ₁ mod (u ⁸ ₁ ) so the image of the homomorphism

H ⁰ (G 24 ; F ₉ ((u 1 )) ^× ) → H ⁰ (G 24 ; Z ) ∼ = Z

is 6 Z , and the result follows.

Note that η is not defined on U 1 (as for example ω ∗ u/u = ωu 6∈ U 1 ), but 8η is well defined.

Proposition 5.2. E ^0,1 ₁ ∼ = H ¹ (G 24 ; U 1 ) ∼ = Z /3 generated by the restriction of 8η.

Proof. The short exact sequence (8) induces a long exact sequence

→ H ¹ (G 24 ; U 1 ) → H ¹ (G 24 ; F ₉ [[u 1 ]] ^× ) → H ¹ (G 24 ; F ^×

9 ) →

The group in the middle is isomorphic to Z /6 by Proposition 5.1 and the group on the right is 2-torsion. The group U 1 is 3-profinite, so there is no 2-torsion in H ^∗ (G 24 ; U 1 ). As H ⁰ (G 24 ; F ^×

9 ) is 2-torsion, the first morphism above is injective.

For the second part of the proposition we use the resolution P _• ^′ of G 24 constucted in Lemma 2.2, to show that the cocycle of 8η can not be a cobord. Recall that η was defined as a crossed homomorphism and thus it can be easily described using the standard (bar) resolution (cf. [1, III.3]). A cocycle representing the image of 8η in the standard resolution B • of G 24 is given by B 1 → U 1 : [g] 7→ g ∗ u ⁸ /u ⁸ . By comparing these resolutions we find a representing cocycle in P _• ^′ . A homomorphism φ • : P _• ^′ → B • over the identity of Z ₃ is given by

φ 0 : P ₀ ^′ → B 0 φ 1 : P ₁ ^′ → B 1

e ^′ ₀ 7→ 1 8

X

g∈Q

g e ^′ ₁ 7→ 1 8 ( X

g∈Q

χ(g ⁻¹ )g)a[a]

where e ^′ _i are the generators e ⊗ 1 of P _i ^′ and {[g]} g∈G

is a G 24 -basis of B 1 (cf. [1, I.5]). Thus the composition

P ₁ ^′ → B 1 → U 1 : e ^′ ₁ 7→ a ² _∗ u ⁸ a ∗ u ⁸ is the desired cocycle. Using the formula from 2.5 we obtain

a ² _∗ u ⁸

a ∗ u ⁸ ≡ 1 − (1 + ω ² )u 1 mod (u ² ₁ ) .

Now we will show that this cocycle can not be a cobord. A morphism from P ₀ ^′ → U 1

sends e ^′ ₀ to a Q 8 -invariant element h of U 1 . Thus by Proposition 5.3 we have h ≡ 1 mod (u ² ₁ ) and thus the composition P ₁ ^′ → P ₀ ^′ → U 1 sends e ^′ ₁ to an element congruent

to 1 modulo u ² ₁ which is not the case for 8η.

5.1. The 0-th line. In the following proposition we give the structure of the 0-th line of the first page of our spectral sequence. We end up with a nice description of the corresponding groups as products of copies of the 3-adics.

Recall that v 1 = u 1 u ⁻² is in degree 4 and ∆ is in degree 24. Thus v ₁ ⁶ ∆ ⁻¹ is in degree 0.

Proposition 5.3.

a) E ^0,0 ₁ ∼ = {g ∈ ((E 2 ) 0 /3) ^G

∼ = F ₃ [[v ₁ ⁶ ∆ ⁻¹ ]] ^× | g ≡ 1 mod (u 1 )} . b) Let g k ∈ E ^0,0 ₁ be such that g k ≡ 1 + v ₁ ^6k ∆ ^−k mod (u ^6k+2 ₁ ). Then

E ^0,0 ₁ ∼ = Y

k≥1

Z ₃ {g k }

k6≡0 mod (3)

c) E ^1,0 ₁ = {h ∈ U 1 | ∃k ∈ (E 2 /3) ^Q ₀

∼ = F ₃ [[ω ² u ² ₁ ]], h = ω ∗ k/k}

d) Let h k ∈ E ^1,0 ₁ be such that h k ≡ 1 + ω ² u ^4k+2 ₁ mod (u ^4k+4 ₁ ). Then E ^1,0 ₁ ∼ = E ^2,0 ₁ ∼ = Y

k≥0

Z ₃ {h k }

k6≡1 mod (3)

Proof. Proposition 2.3 implies (a). By (a) each element g ∈ E ^0,0 ₁ can be written as a product Q

k≥1 g _k ^λ

with λ k ∈ {−1, 0, 1}. As g 3k ≡ g _k ³ mod (u ^18k+2 ₁ ) we obtain the result. To get the action of Q 8 on (E 2 /3) 0 we use formulae (2) and (3) and then (c) follows. As ω ² = −ω ⁶ we have h ⁻¹ _3k+1 ≡ 1 + ω ⁶ u ^4(3k+1)+2 ₁ ≡ (1 + ω ² u ^4k+2 ₁ ) ³ ≡ h ³ _k

mod (u ^12k+8 ₁ ) and we obtain (d).

5.2. The goal of what follows is to construct families of generators {g k } k≥1,k6≡0 mod (3)

and {h k } k≥0,k6≡1 mod (3) as in the previous proposition on which d 1 is easy to de- scribe.

We start with a particular element m ∈ E ^1,0 ₁ that is related to 8η and plays the same role as the element b 1 in [5].

Proposition 5.4. There exists m ∈ E ^1,0 ₁ such that a) m ≡ 1 − ω ² u ² ₁ mod (u ⁴ ₁ )

b) d 1 (m) = 1

c) 24η = m in H ¹ ( G ¹ ₂ ; U 1 ) .

Proof. We imitate the proof of [5, Prop. 5.5] and use paragraph 2.7. By definition 8η is a permanent cycle so there are cochains c ∈ Hom

[[G

]] (P 1 , U 1 ) and d ∈ Hom

[[G

]] (Q 0 , U 1 ) such that c + d is a cocycle in the total complex of the double complex Hom

[[

2

]] (T •• , U 1 ) and such that c represents the restriction of 8η in H ¹ (G 24 ; U 1 ). From the proof of the Proposition 5.2 we have an explicit cocycle c 1 for 8η in the resolution P • , so there exists h c ∈ Hom

[[

2

]] (P 0 ; U 1 ) such that c = c 1 + δ P (h c ). As 24η = 1 in H ¹ (G 24 ; U 1 ) (Proposition 5.2) there exists h ∈ Hom

[[

2

]] (P 0 ; U 1 ) such that δ P (h) = 3c 1 . In the double complex the cochain 3c + 3d is cohomologous to 3d − δ Q (h + 3h c ). One can easily check that h = u ²⁴ /(u ⁸ (a ∗ u ⁸ )(a ² _∗ u ⁸ )). Then 3d − (e − ω) ∗ (h + 3h c ) is a cocycle concentrated in Hom

[[

2

]] (T 1,0 , U 1 ) representing 24η. The formula for m can be deduced using the formulae for the action of a and ω given in paragraph 2.5.

The next lemma is elementary but crutial as it relates the differentials of E and E, and thus suggests that we could use the generators from Theorem 2.4 to construct convenient generators g k and h k .

Lemma 5.5. Let f ∈ F ₉ [[u 1 ]] be such that f ≡ 0 mod (u ^k ₁ ). Then 1

1 + f ≡ 1 − f mod (u ^2k ₁ ) .

Proposition 5.6. Let g k := 1 + v ₁ ^6k ∆ −k for k ≥ 1, k 6≡ 0 mod (3). Then d 1 (g k ) ≡

1 + (−1) ^m+1 ω ² (u ^12m+6 ₁ + u ^12m+10 ₁ ) mod (u ^12m+12 ₁ ) k = 2m + 1 1 + (−1) ^m+1 mω ² u ^12m+2 ₁ mod (u ^12m+4 ₁ ) k = 2m . Proof. By Lemma 5.5 we have d 1 (1+v ^6k ₁ ∆ −k ) ≡ 1+v ^6k ₁ d 1 (∆ −k ) mod (u ^12k ₁ ) where we have used the v 1 -linearity of d 1 . Then the result follows from the formulae from

Theorem 2.4.

5.3. As in [5] we will use the image of d 1 to define h k . We allow the generators h k

to be congruent to 1 − ω ² u ^4k+2 ₁ modulo higher powers of u 1 . Note that 12m + 2 = 4 · 3m + 2 thus for m ≥ 1 and m 6≡ 0 mod (3) we can define

h 3m := d 1 (g 2m )

and then by definition these elements are in the kernel of d 1 .

For the other case, note that 1 + ω ² (u ^12m+6 ₁ + u ^12m+10 ₁ ) ≡ (1 + ω ² u ^12m+6 ₁ )(1 + ω ² u ^12m+10 ₁ ) ≡ (1 − ω ² u ^4m+2 ₁ ) ³ (1 + ω ² u ^12m+10 ₁ ) mod (u ^12m+12 ₁ ). Thus if h m is already definied we could use d 1 (g 1+2m ) to define h 3m+2 by

h 3m+2 := d 1 (g 1+2m )h ³ _m .

As h m ≡ 1 ± ω ² u ^4m+2 ₁ ≡ 1 ± v ₁ ^4m+2 b −1−2m mod (u ^4m+6 ₁ ) we can use Lemma 5.5 to calculate d 1 . In order to do that we will use the partition of N introduced in Theorem 2.4. If −1 − 2m = 1 + 2(3l + 1) then 1 + 2m is divisible by 3. The other cases are of relevance : if −1 −2m = 1 + 2· 3 ⁿ (3l −1) then m = 3 ⁿ (−3l + 1)−1 with l 6≡ 0 mod (3), if −1 −2m = 1 + 2 ·3 ⁿ (9l + 8) then m = 3 ⁿ (−9l −8) −1 with l ≤ −1 and the last one, if −1 − 2m = 1 + 2 · 3 ⁿ⁺¹ (3l + 1) then m = 3 ⁿ⁺¹ (−3l − 1) − 1 with l ≤ −1 . In each of these cases , if h m is defined then h 3m+2 belongs to the same family with n augmented by one. This allows to define recursevely, for l, n ≥ 0 :

h 3

(3l+1)−1 := h ³ ₃

_(3l+1)−1 d 1 (g 1+2·3

(3l+1) ) l 6≡ 0 mod (3) h ₃

_(9l+1)−1 := h ³ ₃

_(9l+1)−1 d 1 (g 1+2·3

(9l+1) )

h 3

(3l+2)−1 := h ³ ₃

(3l+2)−1 d 1 (g 1+2·3

(3l+2) ) .

For this to work we need to define the corresponding element of each family for n = 0. In the first case h 3l is already defined. In the other cases we define:

h 0 := m

h 9l := ω ∗ (1 + v ₁ ^36l+2 b −1−18l )/(1 + v ^36l+2 ₁ b −1−18l ) l > 0 h 9l+5 := ω ∗ (1 + v ₁ ^36l+22 b −11−18l )/(1 + v ^36l+22 ₁ b −11−18l ) l ≥ 0 .

Note that b −1−18l and b −11−18l belong to the two families of Theorem 2.4 that have non trivial image under d 1 . In both of these cases we can apply Lemma 5.4.

This would not have been the case if we would have defined h 0 as 1 + v ₁ ² b −1 as then Lemma 5.5 does not give the enough precision. The following proposition and the recursive definition of the generators describe d 1 : E ^1,0 ₁ → E ^2,0 ₁ . The proof uses Theorem 2.4 and Lemma 5.5.

Proposition 5.7.

z 1,l := d 1 (h 9l ) ≡ 1 + ω ² u ^36l+14 ₁ mod (u ^36l+18 ₁ ) z 2,l := d 1 (h 9l+5 ) ≡ 1 + ω ² u ^36l+30 ₁ mod (u ^36l+34 ₁ ) .

A more complete description of d 1 is given with the following proposition.

Proposition 5.8. The following complexes are exact : Y

n≥0

Z ₃ {g 1+2·3

(3l+1) } → Y

n≥0

Z ₃ {h 3

(3l+1)−1 } → 1 for l 6∈ 3 N Y

n≥0

Z ₃ {g 1+2·3

(9l+1) } → Y

n≥0

Z ₃ {h 3

(9l+1)−1 } → Z ₃ {z 1,l } for l > 0 Y

n≥0

Z ₃ {g 1+2·3

(3l+2) } → Y

n≥0

Z ₃ {h 3

(3l+2)−1 } → Z ₃ {z 2,l } for l ≥ 0 .

The first homology of the following complex Y

n≥0

Z ₃ {g 1+2·3

} → Y

n≥0

Z ₃ {h 3

−1 } → 1

is isomorphic to Z ₃ {h 0 }. In each complex the infinite matrix of the first homo- morphism has the following form

₋₃

1 −3 1 −3

···

and when non trivial the infinite matrix of the second morphism has the following form ( ^{1 3 9 27} ··· ).

Proof. The proof is a consequence of Proposition 5.3, Proposition 5.4, the defini- tions of the generators in paragraph 5.3 and Proposition 5.8.

Corollary 5.9. E ^1,0 ₂ ∼ = Z ₃ .

Proof. This is consequence of the previous proposition. Indeed, the relevant part of the 0-th line of the first page of the spectral sequence is the product over l of the four complexes of the previous proposition together with the exact complex

Y

m>0

Z ₃ {g 2m } → Y

m>0

Z ₃ {h 3m } → 1 .

Theorem 5.10. H ¹ ( G ¹

2 ; U 1 ) ∼ = Z ₃ is generated by 8η.

Proof. We only need to resolve the extension problem 0 → E ^1,0 ₂ ∼ = Z ₃ → H ¹ ( G ¹

2 ; U 1 ) → Z /3 Z ∼ = E ^0,1 ₂ → 0 .

But this is immediate due to Proposition 5.4

6. Getting to H ¹ ( G ¹

2 ; W [[u 1 ]] ^× )

In this section we prove the main theorem by using the short exact sequences from Section 4. The element η again plays an important role in the proof.

Proposition 6.1. H ¹ ( G ¹ ₂ ; F ^× ₉ ) ∼ = F ^× ₉ . Proof. There is a short exact sequence

1 → S ₂ ¹ → G ¹

2 → SD 16 → 1 that gives a spectral sequence and as S ₂ ¹ acts trivially on F ^×

9 we have H ^∗ ( G ¹ ₂ ; F ^×

9 ) ∼ = H ^∗ (SD 16 ; F ^×

9 ) . There is another short exact sequence

1 →< ω >→ SD 16 → Gal → 1 (where Gal := Gal( F ₉ / F ₃ )) and thus a spectral sequence

H ^∗ (Gal; H ^∗ (< ω >; F ^×

9 )) ⇒ H ^∗ (SD 16 ; F ^×

9 ).

By using the standard resolution it is easily seen that the group H ^∗ (< ω >; F ^×

9 ) is isomorphic to F ^×

9 in each degree as ω acts trivially on F ^×

9 .

The group H ¹ (< ω >; F ^× ₉ ) is generated by the identity which is Galois invariant thus

H ⁰ (Gal; H ¹ (< ω >; F ^×

9 )) ∼ = F ^×

9

and by Hilbert 90

H ¹ (Gal; H ⁰ (< ω >; F ^×

9 )) = H ¹ (Gal; F ^×

9 ) = 0 .

As the image of η in H ¹ ( G ¹ ₂ ; F ^× ₉ ) ∼ = H ¹ (SD 16 ; F ^× ₉ ) reduces to the identity in the group H ¹ (< ω >; F ^×

9 ), the differential d 2 : H ⁰ (Gal; H ² (< ω >; F ^×

9 )) → H ¹ (Gal; H ⁰ (< ω >; F ^×

9 ))

has to be trivial.

Corollary 6.2. H ¹ ( G ¹

2 ; F ₉ [[u 1 ]] ^× ) ∼ = Z ₃ × F ^×

9 generated by η.

Proof. The short exact sequence (8) induces a long exact sequence

→ H ⁰ ( G ¹ ₂ ; F ^× ₉ ) → H ¹ ( G ¹ ₂ ; U 1 ) → H ¹ ( G ¹ ₂ ; F ₉ [[u 1 ]] ^× ) → H ¹ ( G ¹ ₂ ; F ^× ₉ ) → H ² ( G ¹ ₂ ; U 1 ) . By Theorem 5.10 we have H ¹ ( G ¹ ₂ ; U 1 ) ∼ = Z ₃ and by Proposition 6.1 we have H ¹ ( G ¹ ₂ ; F ^× ₉ ) ∼ = F ^× ₉ . As U 1 is a 3-profinite group, the first and the last homo-

morphisms are trivial.

Proposition 6.3.

a) H ¹ ( G ¹ ₂ ; F ₉ [[u 1 ]]) = 0

b) The group H ¹ ( G ² ₁ ; W [[u 1 ]]) is 3-profinite.

c) H ¹ ( G ¹ ₂ ; W [[u 1 ]]) = 0.

Proof. (a) is direct consequence of [5, Thm. 1.6]. The group W [[u 1 ]] is 3-profinite and there is a resolution of finite type (Lazard) of the trivial G ¹

2 -module Z ₃ and (b) follows. Multiplication by 3 induces a short exact sequence

W [[u 1 ]] −→ ^×3 W [[u 1 ]] −→ F ₉ [[u 1 ]]

which induces a long exact sequence

−→ H ¹ ( G ¹ ₂ ; W [[u 1 ]]) −→ H ¹ ( G ¹ ₂ ; W [[u 1 ]]) −→ H ¹ ( G ¹ ₂ ; F ₉ [[u 1 ]]) −→

From (a) and the long exact sequence above it follows that the homomorphism H ¹ ( G ¹ ₂ ; W [[u 1 ]]) −→ H ¹ ( G ¹ ₂ ; W [[u 1 ]])

is surjective i.e. the group G := H ¹ ( G ¹ ₂ ; W [[u 1 ]]) is 3-divisible. As G is 3-profinite, it is the limit of finite 3-groups G/I n . Thus the homomorphism G/I n → G/I n

induced by the multiplication by 3 is surjective and therefore G is trivial.

Theorem 6.4. H ¹ ( G ¹ ₂ ; W [[u 1 ]] ^× ) ∼ = Z ₃ × F ^× ₉ generated by η.

Proof. We use the long exact sequence in H ¹ ( G ¹ ₂ ; −) induced from the short exact sequence (7). The homomorphism H ¹ ( G ¹ ₂ ; W [[u 1 ]] ^× ) → H ¹ ( G ¹ ₂ ; F ₉ [[u 1 ]] ^× ) is injec- tive by Proposition 6.3 (c) and also surjective as the image of η ∈ H ¹ ( G ¹ ₂ ; W [[u 1 ]] ^× )

is a generator (by Corollary 6.2).

Finally we get to the proof of 1.6(a).

Theorem 6.5. H ¹ ( G ₂ ; W [[u 1 ]] ^× ) ∼ = Z ²

3 × F ^×

9 generated by η and det.

Proof. We use the short exact sequence (6). We have

H ¹ ( Z ₃ ; H ⁰ ( G ¹ ₂ ; W [[u 1 ]] ^× )) ∼ = H ¹ ( Z ₃ ; Z ₃ ) ∼ = Z ₃ and

H ⁰ ( Z ₃ ; H ¹ ( G ¹

2 ; W [[u 1 ]] ^× )) ∼ = H ⁰ ( Z ₃ ; Z ₃ × F ^×

9 ) ∼ = Z ₃ × F ^×

9 .

The theorem follows from

1 → H ¹ ( Z ₃ ; H ⁰ ( G ¹ ₂ ; W [[u 1 ]] ^× )) → H ¹ ( G ₂ ; W [[u 1 ]] ^× ) → H ⁰ ( Z ₃ ; H ¹ ( G ¹ ₂ ; W [[u 1 ]] ^× )) → 1 Note the det is the image of the identity and the image of η is a generator.

7. P ic ^alg ₂

In this short section we prove of Theorem 1.6 i.e. we calculate P ic ^alg ₂ . We are left with the short exact sequence

0 → P ic ^alg,0 ₂ → P ic ^alg ₂ → Z /2 → 0

that comes from the definition of P ic ^alg,0 ₂ (cf. paragraph 1.1). Note that the isomorphism of Proposition 1.2 sends (E 2 ) ∗ S ² to η (cf. paragraph 3.1). Thus (E 2 ) ∗ S ² is an element of P ic ^alg,0 ₁ that generates Z ₃ × Z /8 in P ic ^alg ₂ . But (E 2 ) ∗ S ¹ is not an element of P ic ^alg,0 ₂ , therefore its image in the above sequence is a generator of Z /2. Thus (E 2 ) ∗ S ¹ itself must generate Z ₃ × Z /16.

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